(FDM) process parameter prediction and optimization using group ...

Int J Adv Manuf Technol DOI 10.1007/s00170-014-5835-2

ORIGINAL ARTICLE

Fused deposition modelling (FDM) process parameter prediction and optimization using group method for data handling (GMDH) and differential evolution (DE) Farzad Rayegani & Godfrey C. Onwubolu

Received: 4 December 2013 / Accepted: 1 April 2014 # Springer-Verlag London 2014

Abstract This paper presents the research done to determine the functional relationship between process parameters and tensile strength for the fused deposition modelling (FDM) process using the group method for data modelling for prediction purposes. An initial test was carried out to determine whether part orientation and raster angle variations affect the tensile strength. It was found that both process parameters affect tensile strength response. Further experimentations were carried out in which the process parameters considered were part orientation, raster angle, raster width and air gap. The process parameters and the experimental results were submitted to the group method of data handling (GMDH), resulting in predicted output, in which the predicted output values were found to correlate very closely with the measured values. Using differential evolution (DE), optimal process parameters have been found to achieve good strength simultaneously for the response. The mathematical model of the response of the tensile strength with respect to the process parameters comprising part orientation, raster angle, raster width and air gap has been developed based on GMDH, and it has been found that the functionality of the additive manufacturing part produced is improved by optimizing the process parameters. The results obtained are very promising, and hence, the approach presented in this paper has practical application for the design and manufacture of parts using additive manufacturing technologies.

F. Rayegani : G. C. Onwubolu (*) School of Mechanical and Electrical Engineering and Technology, Faculty of Applied Science and Technology, Sheridan Institute of Technology and Advanced Learning, Brampton, ON L6Y 5H9, Canada e-mail: [email protected] F. Rayegani e-mail: [email protected]

Keywords Fused deposition modelling (FDM) . Strength . Inductive modelling . Design of experiment (DOE) . Optimization . Group method of data handling (GMDH) . Differential evolution (DE)

1 Introduction Fused deposition modelling (FDM) is a fast-growing rapid prototyping (RP) technology due to its ability to build functional parts having complex geometrical shapes in reasonable build time. Reduction of product development cycle time is a major concern in industries to remain competitive in the marketplace, and hence, focus has shifted from traditional product development methodology to rapid fabrication techniques such as RP [1–5]. Although RP is an efficient technology, full-scale application has not gained much attention because of compatibility of presently available materials with RP technologies [6, 7]. To overcome this limitation, one approach may be the development of new materials having superior characteristics than conventional materials and its compatibility with technology. Another convenient approach may be suitably adjusting the process parameters during fabrication stage so that properties may improve [8, 9]. A critical review of literature suggests that properties of RP parts are the function of various process-related parameters and can be significantly improved with proper adjustment. Since mechanical properties are important for functional parts, it is absolutely essential to study the influence of various process parameters on mechanical properties so that improvement can be made through selection of the best settings. The present study focuses on assessment of mechanical properties viz. tensile, flexural and impact strength of the part fabricated using FDM technology. Since the relation between a particular mechanical property and process parameters related to it is difficult to establish, an attempt has been made to derive the

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empirical model between the processing parameters and mechanical properties using the group method of data handling (GMDH) which is one of the most robust existing inductive modelling methods. FDM is one of the RP processes that build part of any geometry by sequential deposition of material on a layer by layer basis. The process uses heated thermoplastic filaments which are extruded from the tip of the nozzle in a prescribed manner in a semi-molten state and solidify at chamber temperature. The properties of built parts depend on the settings of various process parameters fixed at the time of fabrication. Additive manufacturing (AM), rapid tooling (RT) and reverse engineering (RE) are new manufacturing technologies driven by CAD that make it possible for companies to significantly cut design and manufacturing cycle times. Producing aesthetically appealing AM products that have complex shapes is not difficult. The challenge is to produce manufactured AM parts that are functionally reliable. Therefore, this paper reports the work that has been done to investigate the functionality of manufactured AM parts. The motivation for this emphasis is that no company will want to make a commitment on a technology that may produce aesthetically appealing products that are not functionally reliable. Functionality is therefore a key issue to investigate as this new manufacturing technology is entering into the new phase of its product being adoption as end product rather than a rapid prototype (hence the terminology rapid prototyping).

2 Literature review Recently, research interests have increased amongst researchers to study the effect of various process parameters on responses expressed in terms of properties of built parts. Studies have concluded through the design of experiment (DOE) approach that process parameters such as layer thickness, raster angle and air gap significantly influence the responses of the FDM ABS (acrylonitrile butadiene styrene) prototype [10, 11]. Lee et al. [12]

Fig. 1 Height of slices or layout of layer thickness. Source: [17]

Fig. 2 Orientation of part. Source: [10]

performed experiments on cylindrical parts made from three RP processes such as FDM, 3D printer and nanocomposite deposition (NCDS) to study the effect of build direction on the compressive strength. In their study, out of three RP technologies, parts built by NCDS are found to be severely affected by the build direction. Wang et al. [13] have recommended that material used for part fabrication must have a lower glass transition temperature and linear shrinkage rate because the extruded material is cooled from glass transition temperature to chamber temperature resulting in the development of inner stresses responsible for the appearance of inter- and intra-layer deformation in the form of cracking, de-lamination or even part fabrication failure. Bellehumeur et al. [14] have experimentally demonstrated that the bond quality between adjacent filaments depends on the envelope temperature and variations in the convective conditions within the building part while testing the flexural strength of the specimen. Temperature profiles show that the temperature at bottom layers rises above the glass transition temperature and rapidly decreases in the direction of movement of the extrusion head. Microphotographs indicate that the diffusion phenomenon is more prominent for adjacent filaments in bottom layers as compared to upper layers. Simulation of the FDM process using finite element analysis (FEA) shows that the distortion of parts is

Fig. 3 Raster angle parameter. Source: [10]

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3 DOE and experimentation setup

Fig. 4 Raster width parameter. Source: [10]

mainly caused by the accumulation of residual stresses at the bottom surface of the part during fabrication [15]. The literature reveals that properties are sensitive to the processing parameters because the parameters affect the meso-structure and fibre-to-fibre bond strength. Moreover, it is found that uneven heating and cooling cycles due to the inherent nature of the FDM build methodology results in stress accumulation in the built part resulting in distortion which is primarily responsible for weak bonding and thus affects the strength. Panda et al. [16] performed experiments on the impact of process parameters on dimensional accuracy, surface roughness and mechanical strengths, using response surface methodology (RSM) for modelling and bacterial foraging for finding optimal process parameter settings and responses. From the literature, it is found that a good amount of work has been done in FDM strength modelling; however, little amount of work has been done to develop the strength model in terms of FDM process parameters for prediction purposes. The present study reported in this paper uses the GMDH modelling approach to derive the required relationship among respective process parameters and tensile strength. The models derived serve as predictive models which can be used to anticipate the theoretical best parameter settings that would result in optimal response characteristics. The predictive models are therefore the objective functions while the lower and upper bounds used from the DOE are the constraints, so that the problem becomes constrained optimization which can be solved using any of the existing optimization techniques. In our case, we used differential evolution (DE) to solve the optimization problem. The solutions give the optimal tensile response and optimal process parameter settings.

Fig. 5 Air gap application. Source: [10]

In this study, four important process parameters such as part orientation (A), raster angle (B), raster width (C) and air gap (D) have been considered to study their effects on tensile strength (United Testing System (UTS)). ABS material is used, i.e. the material is constant. The temperature is also considered constant. The definitions of the FDM variable parameters in this study are as follows: (A) The layer thickness which is recognized as the height of the deposited slice from the FDM nozzle as shown in Fig. 1. The layer thickness parameter is used to examine the influence of building thicker or thinner layers on the outcome quality. (B) The orientation of the part is defined as how the part should be positioned when produced as shown in Fig. 2. (C) Raster angle or orientation which is measured from the X-axis on the bottom part layer as shown in Fig. 3. Also, it refers to the direction of the beads of material (roads) relative to the loading of the part. The deposited roads can be built at different angles to fill the interior part. (D) The raster width or road width which refers to the width of the deposition path related to tip size. It also refers to the tool path width of the raster pattern used to fill interior regions of the part curves as shown in Fig. 4. Narrow and wide filling patterns (roads) were considered to be examined. (E) The air gap parameter which is defined as the space between the beads of deposited FDM material as shown in Fig. 5. Hence, the influence of applying positive and negative gaps between the deposited beads was investigated.

3.1 Experimental procedure The 3D models of the specimen are modelled in SolidWorks and exported as an STL file. The STL file is imported to the FDM software (Insight). All testing specimens were constructed in a Stratasys FDM Fortus 400mc System (Fig. 6) in the Advanced Manufacturing and Robotics laboratory at Sheridan Institute of Technology. The laboratory is equipped

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Fig. 8 Thirty-two parts produced for initial testing

Fig. 6 Fortus 400mc System used for manufacturing parts

with both Fortus 400mc and 900mc systems. The tensile test was performed using the UTS, Model SSTM, Serial 1210555, with capacity of 20 kN in accordance with ISO R527:1966 and ISO R178:1975, respectively. 3.2 Pre-test An initial objective was to experiment with the part orientation and raster angle in order to determine how different levels affect the FDM part. The material used is ABS. The different part orientations and raster angles used for this experiment are as follows: Part orientation: {0°, 39°, 90°} Raster angle: {0°, 30°, 45°} for 39 inclined parts In 0° part orientation (Fig. 7a), layered specimens were all fabricated in a build direction that aligned the maximum part dimension with the x-direction of the machine. In 39° part orientation (Fig. 7b), layered specimens were all fabricated in a build direction that aligned the maximum part dimension with the axis that makes an angle of 39° to the x-direction of the machine. In 90° part orientation (Fig. 7c), layered

specimens were all fabricated in a build direction that aligned the maximum part dimension with the y-direction of the machine. Thirty-two parts were produced as shown in Fig. 8 for the initial test. 3.2.1 Part orientation 0° part orientation: {13, 14, 15, 16, 17, 18, 19, 27, 28} 39° part orientation: {20, 21, 22, 23, 24, 25, 26, 29, 30} 90° part orientation: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 32} The averages for the three orientations were used for Fig. 9. As observed, there is no outright linearity for the tensile strength measured for the three orientations. 3.2.2 Raster angle analysis 39° part orientation; 0° raster angle: {22, 25, 29} 39° part orientation; 30° raster angle: {20, 23, 24} 39° part orientation; 39° raster angle: {21, 26, 30} The averages for the three orientations were used for Fig. 10. As observed, there is no outright linearity for the tensile strength measured for the three orientations. 3.3 Full experimental

Tensile strength (MPa)

In this work, factors as shown Table 1 are set as per experiment plan (Table 2) using the DOE methodology found in [18]. All tests are carried out at the temperature 23±2 °C and relative humidity 50±5 % as per ISO R291:1977. The engineering material used for test specimen fabrication is ABS P400. The specimens are fabricated using an FDM 400mc

Fig. 7 a, b, c Part orientations

Part orientation 32 30 28 Part orientation

26 24 22

1

2

3

Fig. 9 Part orientation (1 90°, 2 39°, 3 0°)

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Tensile strength (MPa)

Raster angle 27.00 26.00 25.00 Raster angle

24.00 23.00 22.00

1

2

3

end products. “Overfill” leads to flawed products since the surfaces of the end products have roughages which make the products unpleasant to see although they will have the mechanical strengths expected. From our investigations, we eventually used a −0.0001″ air gap, which gave very good results. It is important to note this “overfill” flaw that might result when we use a negative air gap.

Fig. 10 Part orientation (1 0°, 2 30°, 3 45°)

machine for respective strength measurement. The main FDM variable parameters are considered in this research in Table 1 to evaluate the correlation between these parameters and the proposed response characteristics. The rationale for considering the four variable process parameters for experimentation is given here. The layer thickness is known to affect the AM end product because the smaller the layer thickness, the stronger the finished AM part will be when subjected to axial load. Part orientation is important because when the part is built inclined, it will have the tendency to withstand greater loading in the x-direction and ydirection. The raster angle will have the tendency to affect the internal structure of the finished product. Each neighbouring layer has a raster angle perpendicular to the immediate preceding layer. The raster width is known to affect the finished AM part in such a way that the smaller the raster angle, the greater the tendency for the finished part to withstand higher tensile stress. A negative air gap is known to produce a finished AM part that withstands higher tensile stress. A number of researchers have considered negative air gap and reached these conclusions in their studies similar to our hypothesis [19]. Using the DOE approach, the 16 full factorial conditions were generated as shown in Table 2 for the experimental runs. Each run in the design consists of a combination of FDM parameter levels, and each run result will contain the response of the tensile strength (UTS) measured in megapascals.

3.4 Experimental observations It was observed that the −0.001″ air gap did not give good product results for the −0.008″ raster width since this implies 12.5 % overlap between roads. For this air gap and raster width, we found that there were “overfill” flaws in the AM Table 1 Variable process parameters and their selected low and high levels Variable parameter

Unit

Low level (−1)

High level (+)

Part orientation (A) Raster angle (B) Raster width (C) Air gap (D)

Degree Degree in./mm in./mm

0° 0° 0.008″/(0.2034) −0.0001″/(−0.0025)

90° 45° 0.022″/(0.5588) 0.022″/(0.5588)

4 GMDH The framework for modelling chosen for this applied research is based on the GMDH introduced by Ivakhnenko (details are found in [20–22]) as a means of identifying non-linear relations between input and output variables. The multi-layered iteration (MIA) network is one of the variants of GMDH. The MIA relationship between the inputs and the output of a multiple-input single-output self-organizing network can be represented by an infinite Volterra-Kolmogorov-Gabor (VKG) polynomial of the form [20–22]: yn ¼ a0 þ

M X i¼1

a i xi þ

M X M X i¼1

j¼1

aij xi x j þ

M X M X M X i¼1

aijk xi x j xk …

j¼1 k¼1

where X=(x1,x2,…,xM) is the vector of input variables and A=(a0,ai,aij, aijk …) is the vector of coefficients or weights. When the GMDH network is completed, there is a set of original inputs that filtered through the layers to the optimal output node. This is the computational network that is to be used in computing predictions (in our application, classifications are implied). The best nodes in the input layer (starred nodes in Fig. 11) are retained and form the input to the next layer. The inputs for layer 1 are formed by taking all combinations of the surviving output approximations from the input layer nodes. It is seen that at each layer the order of the polynomial approximation is increased by two. The layer 2 best nodes for approximating the system output are retained and form the layer 3 inputs. This process is repeated until the current layer’s best approximation is inferior to the previous layer’s best approximation. The GMDH network model is constructed during the learning process by the following five procedures (see [23] for details): Step 1 Separating the original data into the training and test sets. The original dataset is separated into the training and testing sets. Training data are used for the estimation of the partial descriptions which describe the partial characteristics of the non-linear system, and the test data are used for organizing the complete description which describes the complete characteristics of the non-linear system.

Int J Adv Manuf Technol Table 2 Input process parameters that affect output responses

Run

A [part orientation]

B [raster angle]

C [raster width]

D [air gap]

Measured (MPa) UTS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 0 0 0 0 0 0 0 90 90 90 90 90 90 90

0 0 0 0 45 45 45 45 0 0 0 0 45 45 45

0.2032 0.2032 0.5588 0.5588 0.2032 0.2032 0.5588 0.5588 0.2032 0.2032 0.5588 0.5588 0.2032 0.2032 0.5588

−0.00254 0.5588 −0.00254 0.5588 −0.00254 0.5588 −0.00254 0.5588 −0.00254 0.5588 −0.00254 0.5588 −0.00254 0.5588 −0.00254

34.07 6.14 29.83 10 38.9 4.44 36.03 8.59 23.94 4.29 23.8 9.45 21.51 3.95 18.37

16

90

45

0.5588

0.5588

7.77

Step 2 Generating combinations of the node input variables in each. All combinations of r input variables are generated before learning each layer. The number of m  combinations is c ¼ r where m is the number of input variables and r is the number of inputs for each node (usually set to two according to the basic model introduced in [20]). Step 3 Calculating the optimum partial descriptions. For each combination, the optimum partial descriptions are calculated, e.g. by applying the regression

analysis to the training data (other approaches utilize, e.g. quasi-Newton gradient method, etc.). The output variables yk in the partial descriptions are called or accessed as intermediate variables. Step 4 Selecting the intermediate variables. The ℜ intermediate variables which give the ℜ smallest test errors calculated for the test datasets are selected from the generated intermediate variables yk. Selected ℜ intermediate variables are used in the following iteration as input variables of the next layer, and calculations from procedures 2 to 4 are repeated. GMDH Network

Fig. 11 GMDH forward feed functional network x 1 x

(1) y

(2)* y

(3) y

x 1 x

(1)* y

(2) y

(3) y

x 1 x

(1)* y

(2)* y

(3)* y

x 2 x

(1) y

(2)* y

(3) y

x 2 x

(1)* y

(2)* y

(3) y

x 3 x

(1)* y

(2) y

(3) y

2

3

4

3

4

4

Input

1

2

3

4

5

6

Layer 1

1

2

3

4

5

6

Layer 2

1

2

3

4

5

6

Layer 3

Output

Int J Adv Manuf Technol Table 3 Input process parameters, measured tensile strength and GMDH predictive model Run

A [part orientation]

B [raster angle]

C [raster width]

D [air gap]

Measured (MPa) UTS

Predicted (MPa) UTS

1 2 3 4 5 6 7 8 9

0 0 0 0 0 0 0 0 90

0 0 0 0 45 45 45 45 0

0.2032 0.2032 0.5588 0.5588 0.2032 0.2032 0.5588 0.5588 0.2032

−0.00254 0.5588 −0.00254 0.5588 −0.00254 0.5588 −0.00254 0.5588 −0.00254

34.07 6.14 29.83 10 38.9 4.44 36.03 8.59 23.94

37.0406 5.190564 31.19316 10.52063 36.58956 5.598833 34.07845 7.896735 22.091

10 11 12 13 14 15 16

90 90 90 90 90 90 90

0 0 0 45 45 45 45

0.2032 0.5588 0.5588 0.2032 0.2032 0.5588 0.5588

0.5588 −0.00254 0.5588 −0.00254 0.5588 −0.00254 0.5588

4.29 23.8 9.45 21.51 3.95 18.37 7.77

4.123696 21.50698 9.825788 22.04529 4.560469 21.80001 7.018248

Step 5 Considering stopping the multi-layered iterative computation. When the error of the test data predictions resulting from the last layer stops decreasing, the iterative computation is terminated. The complete description of the characteristics of the non-linear system can be constructed by using the optimum partial descriptions generated in each layer in the form of a network. It could be summarized that the GMDH-type polynomial networks influence the contemporary artificial Fig. 12 Surface plot of the process parameters for runs 13–16

neural network algorithms with several other advantages [20–24]: 1. They offer adaptive network representations that can be tailored to the given task. 2. They learn the weights rapidly in a single step by standard ordinary least square (OLS) fitting which eliminates the need to search for their values and which guarantees finding locally good weights due to the reliability of the fitting technique.

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3. These polynomial networks feature sparse connectivity which means that the best discovered networks can be trained fast. Some limitations of the GMDH technique include selection of input arguments, inaccuracies in parameter estimation, multicollinearity, reduction of complexity, formulas of partial descriptions, overfitting, partition of data and low accuracy.

y ¼ 33:8477 þ 0x1 þ 0x2 −1:6447x3 −45:5050x4 − 0:0011 x1 x2 þ 0:0206 x1 x3 þ 0:2293 x1 x4 − 0:0671 x2 x3 − 0:0732 x2 x4 þ 33:1809 x3 x4 −0:0013x21 þ 0:0020x22 −4:6791x23 −23:8441x24

ð1Þ

where x 1 = A (part orientation), x 2 = B (raster angle), x3 =C (raster width) and x4 =D (air gap). 4.2 Quality of the hybrid GMDH network results

4.1 Hybrid GMDH network Based on the shortcomings of the basic GMDH, hybrids of GMDH were proposed to significantly enhance the performance of GMDH [22]. The framework for modelling chosen for this applied research work is based on hybrid GMDH (details are found in [23]). The GMDH predictive model for the part produced by the Stratasys FDM 400mc System is given in Table 3. The GMDH model for the part produced by the Stratasys FDM 400mc System based on Table 3 is

Fig. 13 Pseudocode for classical differential evolution

It is necessary to assess the quality of the results obtained from the GMDH network. In order to do this, some statistical data are given here. First, using the first 12 predictions for training and the last 4 predictions for testing in Table 2, it is found that the training error is 1.22 % and the testing error is 1.4 %. The sum of square residual, SS: residual: J=0.1882; the sum of square deviation, SS: deviation from mean: S=5.8697e+03; hence, the coefficient of determination given as 1−J/S is given as: r2 =1.0000. The regression sum of square, SS_regression= 1.6900e−10; the error sum of square, SS_error=5.0487e−27; the mean square regression, MSE_regression=3.3800e−11;

Input: Populationsize , Problem size , Weighting factor , Crossoverrate Output: S best Population InitializePopulation( Populationsize , Problem size ) EvaluatePopulation(Population) S best GetBestSolution(Population) While ( StopCondition()) NewPopulation For ( Pi Population) NewSample( Pi , Population, Problem size , Weighting factor , Crossoverrate ) Si If (Cost( S i Cost( Pi )) NewPopulation S i Else NewPopulation Pi End End Population NewPopulation EvaluatePopulation(Population) S best GetBestSolution(Population) End Return ( S best ) Figure 13 Pseudocode for classical Differential Evolution The overall steps involved in the classical DE are summarized here: Step 1: Initialization Step 2: Mutation Step 3: Crossover Step 4: Selection Step 5: Stopping criteria

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Tensile Strength

5 Optimization of process parameters for tensile strength prediction 5.1 Mathematical formulation of the tensile strength problem The hybrid GMDH tensile strength model developed is utilized by the continuous DE for optimization in order to determine the optimal combinations of part orientation (A), raster angle (B), raster width (C) and air gap (D) that result in maximizing tensile strength for the FDM part. The optimization problem is essentially that of using Eq. 1, as objective function with constraints taken from Table 2. The tensile strength optimization problem can now be fully mathematically stated as follows: y ¼ 33:8477 þ 0x1 þ 0x2 −1:6447x3 −45:5050x4 −0:0011x1 x2 þ 0:0206x1 x3 þ 0:2293x1 x4 −0:0671x2 x3 −0:0732x2 x4 þ 33:1809x3 x4 −0:0013x21 þ 0:0020x22 −4:6791x23 −23:8441x24 0 ≤ x1 ≤90 0 ≤x2 ≤ 45 0 :2034≤x3 ≤ 0:5588 −0:0025≤x4 ≤ 0:5588

ð2Þ

s:t:

ð3Þ

5.2 Differential evolution scheme The DE algorithm introduced by Storn and Price [25] is a novel parallel direct search method, which utilizes Np parameter vectors as a population for each generation G. DE is one of the extant evolutionary approaches used to solve complex real-life problems. It was primarily designed for continuous domain space formulation but was reformulated to solve permutative problems by [26]. Differential evolution has a specialized nomenclature that describes the adopted configuration. This takes the form of DE/x/y/z, where x represents the solution to be perturbed (such as random or best). y signifies Table 4 DE control parameters used for experimentation

Population size, NP Number of parameters Mutation probability, F Crossover probability, CR Number of generations

50 4 0.20 0.60 1,000

MPa

the mean square error, MSE_error=1.8031e−28; the F-value=1.8745e+17. Figure 12 shows the surface plot of the process parameters for runs 13–16, showing that the modelling search space is highly complex.

45 40 35 30 25 20 15 10 5 0

Measured Predicted

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fig. 14 Measured and predicted tensile strength response

the number of difference vectors used in the perturbation of x, where a difference vector is the difference between two randomly selected although distinct members of the population. Finally, z signifies the recombination operator performed such as bin for binomial and exp for exponential. The algorithm in Fig. 13 provides a pseudocode listing of the differential evolution algorithm [27–29] for optimizing a cost function, specifically a DE/rand/1/bin configuration. The overall steps involved in the classical DE are summarized here: Step 1 Step 2 Step 3 Step 4 Step 5

Initialization Mutation Crossover Selection Stopping criteria

Using the classical DE [25], which is one of the extant evolutionary approaches, the optimal parameter settings for the FDM tensile test are found. DE was used for optimizing the tensile strength optimization problem expressed in Eqs. 2 and 3. The control conditions used for optimization are shown in Table 4. The approach of using the hybrid GMDH model of Eq. 2, as objective functions with constraints given in Eq. 3, for optimizing the tensile strength problem is more straightforward than when ANN is employed in modelling. This type of mathematical formulation makes the GMDH response models more useful to the end user since the models for the problem being solved are transparent and could be used for future applications. Moreover, the mathematical models of Eqs. 2 and 3 are easy to use as the objective functions by most standard optimization techniques for determining optimal process and response conditions. Table 5 Optimal process parameters from DE

Tensile strength Part orientation (A) Raster angle (B) Raster width (C) Air gap (D)

MPa Degree Degree in./mm in./mm

Mean

Standard deviation

36.8604 0° 50° 0.008″/(0.2034) −0.0001″/(−0.0025)

5.64098×10−6 7.3311×10−5 2.16807×10−14 6.78911×10−7 8.8219×10−19

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6 Results and discussions The process parameters A, B, C and D and tensile response of Table 2 were submitted to the GMDH system to develop a predictive model that relates the tensile strength to the process parameters. The last column of Table 3 shows the predicted values obtained from the GMDH modelling system while the second to the last column shows the experimental values. Figure 14 shows the experimental (measured) and predicted values in a graphical form for the 16 runs. As could be observed, our GMDH modelling system predicts very well the behaviour of the tensile strength response with very little deviation. For the experimentation, the simulation was run 30 times with the average running time per simulation being 0.5 s. The mean optimal process parameters and tensile strength as well

Table 6 Details of the DE simulation runs and the relevant statistical data

as the standard deviation that the DE found are given in Table 5. Details of the DE simulation runs and the relevant statistical data (last two rows) are shown in Table 6.

7 Conclusions In this work, a functional relationship between process parameters and tensile strength for the FDM process has been developed using the group method for data modelling for prediction purposes. An initial test was carried out to determine whether part orientation and raster angle variations affect the tensile strength. It was found that both process parameters affect tensile strength response. For more elaborate experimentation, the process parameters considered are layer thickness, orientation,

Run no.

x1 (A)

x2 (B)

x3 (C)

x4 (D)

y (tensile strength)

1 2

0.000127 0.000308

49.9999 49.9999

0.2034 0.2034

−0.0025 −0.0025

36.860376 36.860369

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 18 20

0.00002 0.000001 0.0000113 0.000041 0.00006 0.000019 0.000082 0.000052 0.000166 0.000098 0 0.000026 0.0000284 0.0001 0.000043 0.00002 0.000276 0.00017

49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999

0.203402 0.2034 0.2034 0.203401 0.2034 0.2034 0.2034 0.203401 0.203401 0.203401 0.2034 0.2034 0.2034 0.2034 0.203401 0.203401 0.2034 0.203402

−0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025

36.860373 36.860387 36.860379 36.860376 36.860384 36.860377 36.860384 36.860385 36.860375 36.860376 36.860385 36.860389 36.860375 36.860384 36.860382 36.860386 36.860378 36.860369

21 22 23 24 25 26 27 28 29 30 Mean SDV

0.000052 0.000083 0.000098 0.000113 0.000059 0.000112 0.000046 0.000015 0.000077 0.000086 7.96567e−05 7.3311e−05

49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999 49.9999

0.2034 0.2034 0.2034 0.2034 0.2034 0.2034 0.203402 0.203401 0.2034 0.2034 0.203400433

−0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025 −0.0025

36.860378 36.860387 36.860386 36.860385 36.860379 36.860379 36.860381 36.860389 36.860383 36.860388 36.8603808

2.16807e−14

6.78911e−07

8.8219e−19

5.64098e−06

Int J Adv Manuf Technol

raster angle, raster width and air gap. The process parameters and the experimental results were submitted to GMDH, resulting in predicted output, in which the predicted output values were found to correlate very closely with the measured values. Since the FDM process is a complex one, it is really challenging to determine a good functional relationship between responses and process parameters. Using DE, which is one of the extant evolutionary approaches, optimal parameter settings are found. Our investigations have shown that a negative air gap significantly improves the tensile strength. Smaller raster widths also improve tensile strength. Part orientation plays a major role as could be observed from the results. For zero part orientation (with the part orientation coinciding with the direction of tensile loading), maximum tensile strength is obtained. Increased raster angle also improves tensile strength, although not very significantly. The optimized solutions that DE found agree very reasonably with our observations from Table 2. Our experimental results in Table 2 show that maximum tensile strength is obtained in run 5 for which the part orientation is zero, raster angle is 50°, raster width is 0.2034, with negative air gap of −0.0025. DE optimal solutions match these observations. Consequently, the conclusions reached in this research are reliable and can be used in real-life applications. Acknowledgements Attila Nagy and Andrew Orton, the two Sheridan Centre for Advanced Manufacturing and Design Technology (CAMDT) Laboratory technologists who assisted us with some of the additive manufacturing procedures, are gratefully appreciated.

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